Rev. Acad. Canar. Cienc., XVIII (Núms. 1-2), 101-132 (2006) (publicado en agosto de 2007)
ON THE RECEPTION AND SPREAD OF METAPHYSICAL
EXPLANATIONS OF IMAGINARY NUMBERS IN SPAIN
José M. Pacheco*, Francisco J. Pérez-Fernández** & Carlos Suárez**
Centro de Investigación de Matemática Aplicada (CIMA)
Facultad de Ingeniería. Universidad del Zulia
Apartado 10482, Maracaibo-Venezuela
e-mail: aisolinap@hotmail.com
The introduction in Spain of the ideas on complex quantities is documented through the
work of José María Rey Heredia, a Professor of Logic and self'.-taught mathematician who wrote
during his last life years the book Teoría Transcendental de las Cantidades Imaginarias. Rey was a
follower of Kant and Krause, and tried to inscribe the development of the theory into the
intellectual framework ofTranscendental Logic. His attempt is a lengthy and sometimes erroneous
comment, on the Mémoire on imaginary quantities published by the Abbé Buée in the Philosophical
Transactions the year 1806. Rey had a number of followers who introduced his ideas in teaching
through the second half of the l 9'h Century as an aide to the introduction of various geometrical
concepts.
Resumen
La introducción en España de las ideas filosóficas sobre los números complejos se
encuentran documentadas en la obra de José María Rey Heredia, Profesor de Lógica y matemático,
quien escribió una especie de testamento matemático filosófico durante los últimos años de su vida,
la Teoría Transcendental de las Cantidades imaginarias. Rey fue seguidor de Kant y de Krause e
intentó situar los números complejos en el marco de la Lógica Trascendental kantiana. El texto de
Rey es un largo y desigual comentario, a la luz del esquema de las categorías, de la Memoria
publicada por el Abate Buée en las Philosophical Transactions el año 1806. Rey tuvo varios
seguidores que introdujeron sus ideas en la enseñanza media durante la segunda mitad del XlX
como apoyo al estudio de la Geometría.
AMS Classifications: OlA55, 01A90
Keywords: lmaginary quantities, complex numbers, Kant, Spain, l 9th Century
l. Preliminaries
The introduction of rigour in Mathematics is a topic that can be addressed
from at least two different viewpoints. A first one, mostly encountered in
contemporary Mathematics, stems from the explosive developments which took
place during the l 9th Century, and is represented by the fact that practitioners are
trained to absorb certain techniques amounting to a carefully prepared mixture of
computing capabilities, cross references, and the ability to derive results from well
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authorised sources. As a rule this is considered as the standard and to a certain
extent it has been able to ban most intuitive reasoning pattems in large
mathematical areas. The second viewpoint is the attempt to base mathematical
derivations on sound principies regarding the philosophical status of the ideas
involved. History offers a dearth of nomenclature such as negative, impossible,
surd, and imaginary numbers, to quote just a few examples showing that sorne
operations performed within the usual - for their times- number domains could
yield results without existence in them. Ali the above names suggest sorne sort of
strangeness that should be managed in order to "justify" computations that would
otherwise remain in the realm ofpurely formal ones.
Both standpoints have quite different natures and areas of application.
While the first one is emphasised when a theory is axiomatically developed and
has a very syntactic flavour, the second one is reserved to more foundational
matters and tries to preserve unity of Mathematics by establishing conceptual
connections between succesive enlargements and developments. Here a case study
of the second class is presented, namely an episode related to the introduction of
complex numbers in Spain during the l 9th Century. lt will be shown that rigour in
the second sense can be accompanied by a lack of it in the first sense.
Nevertheless, quite revealing pieces of Philosophy and Mathematics will appear
along this paper.
The philosophical status of negative and imaginary numbers was a
recurrent concern for sorne enlightened philosophers and mathematicians during
the second half of the l 8th Century, even though their use can be tracked in
mathematical practice back to the earlier days of Algebra in the works of Giro lamo
Cardano (1501-1576) and Raffaelle Bombelli (1526-1578) when dealing with the
classical problem of dividing a given number into two numbers whose product is
known. lt must be noted that the interest of these considerations was enhanced due
to the emergence of Algebra as a sort of Universal Arithmetic deeply related with
metaphysical considerations on the validity of the items under manipulation. A
very interesting observation on these precursors is that they considered complex
roots of an equation not as solutions thereof strictu sensu, but in the sense of
complying with the usual symmetric functions of the roots, an idea still used by
191h Century theoreticians, see e.g. [Gilbert 1831]: This is a very early appearance
of the idea of a weak solution, much in the same sense that non-differentiable
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functions can be considered solutions of differential equations within the
framework of Distribution Theory. Moreover, by 1800 Science was coming of age
for the introduction of vector magnitudes, where the mere idea of adding them
amounts to the simultaneous consideration of severa! properties embodied in the
components.
The reception of ali these advances took place in Spain quite late, well into
the l 91h Century, and is found in textbooks used for severa! decades in secondary
teaching, and even at the university leve!, until the first years ofthe 20th Century.
The history of Spanish Mathematics during the l 91h Century, though
possibly not as exciting as that in other European countries -there is no reference to
Spain in [Klein 1926]-, is intimately related with the turbulent political and cultural
atmosphere that prevailed through the Century and was responsible for a variety of
educational and research policies in Spain. A detailed account is still missing,
though an interesting attempt has been recently undertaken in [Suárez 2006].
2. Rey Heredia: a biographical sketch
The history ofthe study of complex numbers in Spain from a philosophical
viewpoint is that of the endeavour of the Andalusian philosopher and self-taught
mathematician José María Rey Heredia, who lived between 1818 and 1861 - a
contemporary of Bernhard Riemann ( 1826-1866)-, and is contained in his
posthumous book Teoría Transcendental de las Cantidades Imaginarias (A
Transcendental Theory of Imaginary Quantities, for short Teoría in what follows)
published the year 1865 after the author's death, at the expenses of the Spanish
Education Ministry [Rey 1865]. The book was presented by Rey's estate to a
national contest for texts with ideas applicable in improving quality of the Spanish
Secondary Education, following the wake of the 1857 Act known as Ley Moyana
after the Minister Claudio Moyano (1809-1890) who prometed it. Most
biographical data on the author are found on the long foreword to Teoría by the
Academician Pedro Felipe Monlau (1808-1871), a politician and physician who
co-authored with him an earlier two-volume book entitled Curso de Psicología y
Lógica published during 1849 in Madrid.
Rey was bom in Córdoba and at the age of fifteen he entered the Seminar,
where he read Philosophy and Theology for nearly eleven years in order to become
a priest. This was a rather common practice at the time for lower class families
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when they happened to have a son with good intellectual aptitude, for children
were educated and taken care of for severa! years with little or no cost to the
parents. He did not receive the orders, but during his last years at the Seminar he
collaborated in teaching younger students matters such as Philosophy and French.
For sure the ecclesiastical build up of Rey influenced his texts, where Latin
quotations are unforgivingly presented without translation. On leaving the Seminar
he was offered a chair at a local school for teaching Logic, but he declined it and
went instead to Ciudad Real to occupy the chair of Logic at an official secondary
school. After obtaining in 1846 the civil degree of Bachiller en Filosojia, he was
appointed Catedrático -the Spanish title for full professor- of Logic at the Instituto
del Noviciado in Madrid, a high school which is still open as such under the name
of Instituto Cardenal Cisneros, where he wrote the Logic volume ofthe Curso and
met in 1850 the mathematician and Professor Acisclo Vallín Bustillo ( 1825-1895),
who introduced him to the world of complex numbers. He went on obtaining
different degrees: Bachiller en Jurisprudencia (1852), Licenciado en
Jurisprudencia (1854), and Licenciado en Filosofia y Letras (1857). In bis prívate
life he was a respectable, not very healthy man who died from tuberculosis, the
same illness from which his young wife had passed away in 1854, and at his early
death a street in Córdoba received his name. Now it is a busy and long street and
very few locals, if any, do know that it wears the name of a mathematician. A
portrait of Rey can be seen in Figure 1.
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Figure 1: The Spanish philosopher and mathematician José María Rey Heredia (1818-1861), and his
signature.
3. Rey's cultural and philosophical background
Rey's Elementos de Lógica (Lógica in what follows, see Figures 2 and 3) is
a text published in 1853 when the two-volume textbook written with Monlau
evolved into two separate books, and is the first source to know about the author's
cultural and philosophical background and his opinions. Lógica had many
succesive editions until the end of the l 91h Century. For this study a copy of the
l 21h editi'on [Rey 1849] has been employed, published by the classical printers
Sucesores de Rivadeneyra in Madrid the year 1883, see Figure 2. The text is a
treatise on classical Logic encompassing Grammar and Dialectics for use at
secondary institutions, and it is aimed to mastering the art of enchaining plausible
clauses and maintaining, and hopefully winning, dialectic challenges. The
exposition shows a remarkable unity and is written in very good Spanish with the
usual l 91h Century flourished style. It is divided into four parts or books and a long
introductory chapter entitled Prenociones where in a schematic though clear way
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the main concepts and ideas are presented. Figure 3 shows the definition of Logic
in the first line ofthis introduction.
SUllABIO
PiDOOIOlUB.
Figures 2 and 3: Lógica , and the definition of Logic, as expressed in Lógica.
From the study of Lógica it is difficult to ascribe Rey to any particular
philosophical current. Indeed, by the time he wrote the book, in his early thirties,
he was aware of various classical problems and of the work of the more known
authors, but he seems not to have made up his mind in any definite sense.
Moreover, andas the 121h edition shows, Rey did not include anything related with
the new trends in Mathematical Logic developed by his contemporary George
Boole (1815-1864), whose An investigation into the Laws of Thought, on which
are founded the Mathematical Theories of Logic and Probabilities was published
in 1854. Nevertheless, in p. 256-257 he states, when dealing with the sorites
or inference through enchained propositions, that :
Puede compararse el sorites á una serie de ecuaciones en que concluimos
la igualdad de dos extremos por ser iguales á varios términos medios ... 1
1 Sorites can be compared with a sequence of equations, where we conclude equality ofthe
extremes through the equality ofthe intermediate terms ...
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Sorne studies [Calero 1994] point out at Rey as a late idéologue, a follower
of the philosophical, linguistic and pedagogical school developed around Antoine
Destutt-Tracy (1754-1836) in the final years ofthe 181h Century on the steps of
John Locke (1632-1704), the Abbé Condillac ( 1715-1780) and others [Picavet
1891]. The group was a rnost influencing one in the design of educative policies in
pre-napoleonic France and had a strong influence in sorne other countries. For
instance, Destutt's book Élements d'Idéologie was the source of the Spanish text
by the mathernatician Juan Justo García (1752-1830) Elementos de Lógica
Verdadera, practically a translation of Destutt [Cuesta 1980]. published in Madrid
the year 1821. García is credited to be arnong the first to introduce Calculus in
Spanish Higher Education [Cuesta 1985]. The influence of this school can be
noticed even in texts such as [Colburn 1831] in North Arnerica:
Success in reasoning depends very much upon the language which is
applied to the subject, anda/so upon the choice of the words that are to
beused(p. 241).
For this study it is quite relevant to quote Rey's définition idéologique of a sign
(or syrnbol) in page 114 of Lógica:
[un signo es una} cosa cualquiera considerada como medio que nos
conduce al conocimiento de otra2,
an idea used later in Teoría when dealing with the various interpretations of the
irnaginary unit i = ~. Maybe the best choice would be to assess Rey as an
eclectic, as shown by the authors cited in Lógica, like Pierre Royer-Collard (1763-
1845) and Victor Cousin (1792-1867) or Felicité Lamennais (1782-1854), by sorne
dubitative expressions on the still prevalent Port Royal Logic - he supports it on
page 124 but is against it in page 118, and by his farniliarity with sorne
philosophers of the Scottish School of Cornrnon Sense, like Thomas Reid (1710-
1796) [Reid 1764].
On reading Lógica, the narne ofEmmanuel Kant (1724-1804) is found only
once in the following phrase (p. 232):
El paso de las proposiciones a sus contrarias las llama Kant
"raciocinio del entendimiento."3,
2 [A sign is} any thing considered as a means towards the knowledge of another thing.
3 According to Kant, the passage from a proposition to its contrary is a "judgement of our
understanding"
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and no mention is made of Transcendental Logic or any other kantian ideas. Thus,
the enthusiastic application -even to the point of including an Appendix containing
a translation of the first part of the Transcendental Logic extracted from the
Critique of Pure Reason- of Kant's methods and concepts made by Rey in Teoría
is a novelty, showing that he must have been among the first to have read Kant,
possibly in the early partial translations made after French versions4 or even
directly in French. See [Albares 1996] for an account of the reception of Kant in
Spain during the first half ofthe l 91h Century. Therefore, it is uncertain how deeply
Rey knew about Kant, though he was later considered and classified as a kantian
[Méndez Bejarano 1927]. Indeed he does not doubt when it comes to apply the
Theory of Categories to the treatrnent of imaginary quantities.
Another striking fact about Rey shown in Teoría, apart from his quoting
Reid, is his knowledge of two British authors counting among the developers of
Algebra in a modern sense: George Peacock (1791-1858) and John Warren (1796-
1852). The main impression is that during the last decade of his life Rey was able
to update his philosophical and mathematical background, quite possibly through
preparation to the succesive exams he took in order to obtain his academic degrees.
4. Rey's Teoría
Teoría is a beautifully printed book (Figure 4) with XX+330 pages
produced the year 1865 by the Imprenta Nacional at Madrid. Rey started working
on it during 1855 and devoted the last five years of his life to writing the book,
whose Introduction was the part he wrote last. Teoría is a piece of very cultivated
Spanish and is illustrated with a large amount of line drawings and lithographies
supervised by Vallín, who also checked fonnulas and cornputations, with little
success in various cases, as shown by a detailed study ofthe book. To Rey, the task
ofthe philosopher / rnathematician is stated in the rnotto attributed to Pierre Sirnon
de Laplace (1749-1824) through F. Coyteux - the author of a book published in
Paris by Moreau in 1845: Exposé d'un systéme philosophique suivi d 'une théorie
des sentiments ou perceptions- he quotes at the very beginning ofthe book:
4 The first substantial direct translation ofthe Critique of Pure Reason from German into Spanish,
although nota complete one, was the work ofthe Spanish-Cuban politician and philosopher José
del Perojo Figueras (1852-1908) and appeared in 1883. There exist severa! editions of this
translation published by Losada in Buenos Aires, the last one dating from 1979. Perojo was elected
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Figure 4. Cover page of Teoría
JI faut refaire les Mathématiques, les placer sur un nouveau piédesta/ : JI
serait a désirer que ce fút l'oeuvre d 'un homme nouveau, qui fút étranger
aux mouvements et aux progres des sciences et n 'en connút que les premiers
é/éments.
This programme, in a certain sense a forerunner of the Jogicist schools around
1900, seems to be at the basics of ali subsequent studies on mathematical
foundations. Of course Rey insists on it severa) times along Teoría, and especially
in the final part of the Introduction where he wrote his most deep belief on this
idea and on the adequacy of the Transcendental Logic to <leal with complex
numbers.
twice to the Congress as a representative ofLas Palmas, where a statue ofhim as well as a street
with his name can be found.
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Rey's Teoría cannot be considered asan isolated attempt. There exist more
books and surveys of his time trying to make clear the role of imaginary quantities
and other sorts of mathematical objects from a foundational viewpoint. In addition
to the earlier Treatise on the geometrical interpretation of the square roots of
negative quantities of 1828 by Warren, there is the long report [Matzka 1851],
quoted in [Hoüel 1874], and the interesting Dutch dissertation [de Haan 1863]. In
the present state of knowledge, it is impossible to assess whether Rey knew about
these two authors, Matzka or de Haan, and most possibly the answer is in the
negative, because Rey's sources are usually French or British writers, although no
mention, not even an indirect one, is found of William Rowan Hamilton (1805-
1865), to quote but one important absence.
5. The French Connection: Rey and Buée, and Pascal
In his very interesting Introduction to Teoría Rey attributes to a certain Mr.
Buée, emigrado francés 5 the glory of discovering or pointing out, that imaginary
quantities can be used to represent perpendicularity.
Adrien-Quentin Buée (1748-1826) was a French priest belonging to the
large group of émigrés, mainly aristocrats and clergymen who fled from France
into England from 1792 onwards as a result of the French Revolution. After that,
by 1814 most ofthem were back in France. Among them there was a number who
were intellectuals welcome by the British leamed people, and Buée is a clear
example. He went to Bath, where he tried to publish in 1799 a Jeaflet titled
Recherches Mathématiques sur la texture intime des corps, though it seems that it
never carne into print. Later on he wrote an article for the Joumal of Natural
Philosophy, Chemistry and the Arts (1804): Out/ines of Mineralogical Systems of
Romé de /'Is/e and the Abbé Haüy, with Observations - by the way, Haüy was
another émigré and haüyne, a bright blue mineral appearing very often in volcanic
lava, is named after him- and in 1806 he published in the Philosophical
Transactions of the Royal Society the memoir on imaginary numbers [Buée 1806]
(Figure 5) that was to inspire Rey's research. It is remarkable that it was published
in French; only very few papers are written in languages other than English in this
Joumal : putting apart the Latín which coexisted with English until well into the
5 Mr. Buée, a French émigré.
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l 81h Century, no more than ten articles were published in French in the whole
history of the Transactions, and from 1810 onwards only English is used. Buée
appears again in Paris taking care of the 1821 new edition of a Dictionnaire des
termes de la Révolution he had anonimously printed in 1792 before escaping to
England. Information on Buée can be found in [Flament 1982].
e ss :i
111. Mémoire sur les Qμantités imaginaires. Par M. Buée.
Communicated by William Morgan, Esq. F. R. S.
Read June SBO, 1805.
Des Signes + et - .
x, CEs signes ont des significations opposées.
Considérés comme signes d'Oj>érations arithmétiques, + et -
sont les signes, l'un de l'aJditi<m, l'autre de la soustrach·on.
Considérés comme signes d'opérations géométriques, ils indiquent
des directions opposüs. Si l'un, par exemple, signifie
qu'une ligne doit etre tirée de gauche a clroite, l'autre signifie
qu'elle doit ctre tirée de droite agauche.
Figure 5: First lines of Buée's memory on imaginary numbers.
Buée's views on imaginary numbers conform one of the last episodes m the
acceptation of negative and imaginary numbers as valid mathematical entities on
metaphysical grounds, a topic that had been dealt with along the previous Century
by authors like Abraham de Moivre (1667-1754), a follower of John Wallis (1616-
1702) [Smith 1959], Jean Argand (1768-1822) or Caspar Wessel6 (1745-1818)
6 In Wessel's book (translation by Zeuthen) the following construction is found: « Désignons par
+ 1 la unité rectiligne positive, par +e une autre unité perpendiculaire a la premiere et ayant la
méme origine: alors l'angle de direction de+ 1 sera égal a Oº, celui de -1a180º, celui de +e a
90°, et celui de -e to -90º ou a 270º; et se/on la regle que l'angle de direction du produit est égal
a la somme de ceux desfacteurs, on aura:(+ 1)(+ 1) = + 1; (+l)(-1)=-l; (-1)(-1)=+ 1; (+ l)(+e)=+e;
(+1)(-e)=-e; (-J)(+e)=-e; (-1)(-e)=+e; (+e)(+e)=-1; (+e)(-e)=+l;(- e)(+e)=-1. 11 en résulteque e est
égal a ~et que la déviation du produit est déterminée de te/le sorte qu 'on ne tombe en
contradiction avec aucune des regles d'opération ordinaires ».
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[Smith 1953, Wessel 1797) on the side of supporters, while one of the strongest
opponents was William Frend (1757-1841) who published his Principies of
Algebra in 1796. The ideas distilled by Buée and Robert Woodhouse (1773-1827)
[Woodhouse 1801 and 1802) were to inspire more than one book, e.g. that of
Warren, another source for Rey, and parts of Peacock's Treatise on Algebra o[
1830. Buée is credited as well in Hoüel's Théorie Élementaire des Quantités
lmaginaires, published in Paris as late as 1874 [Hoüel 1874).
lt is surprising the neopithagoric flavour employed by Rey, insisting on it
along Teoría, emphasising on the single phrase of Blaise Pascal (1623-1662) taken
from Pensées (Série XXV, nº 592 )[Pascal 1977, Vol ll, p 138) and cited five times
in the book:
Los números imitan al espacio, aunque son de naturaleza tan diferente
which he uses as the ultimate authoritative argument in various chapters of Teoría.
The original French reads Les nombres imitent l'espace, qui sont de nature si
différente. A point of interest could be Rey's translation, where the French qui is
translated into the Spanish aunque (though, although). Possibly it is a matter of
overemphasising on the different nature of numbers and space. Rey is aware that
numbers are the mathematical representation of time, as opposed to space, and he
is shocked by the application of numbers to the description of space. Maybe the
preparation of Lógica with its survey of the Port-Royal Logic induced Rey to a
study of Pascal, where he found condensed in a single phrase the Cartesian idea of
translating Geometry into equations which he was to follow and develop in Teoría.
Later on, he must have found it most adequate as a vivid representation of the
application of kantian categories to the study of complex numbers.
6. A survey of Teoría
Teoría is essentially a large and erudite commentary ofBuée's Mémoire o[
1 806, which is closely followed in the introductory parts of the book. From the
mathematical and historical perspective the points of higher interest are found in
the Introduction, in Book I (Chapters 1 and II), Book 11 (Chapter ll), and in Book
IV. The extant chapters are devoted to an exposition of the usual algebraic
algorithms and geometrical interpretations of complex numbers and the most
interesting features are the line figures and wonderful lithographies in white on a
black background, as well as the intricacies of sorne complicated explanations.
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J
6.1. The Introduction
The Introduction is a long programmatic text extending along twenty-two
pages of very fluid and elaborated Spanish. Divided into thirteen sections devoted
to severa! aspects on the general ideas of Mathematics, the notion of imaginary
quantities, their history, the necessity of Metaphysics in Mathematics, sorne
authoritative quotations and phrases with an oratoria] tlavour, it is the most brilliant
part of Teoría and is worth reading and enjoying its phrasing.
The first section deals with the exactness of Mathematics and asserts that in
sorne future no obscurities or mysteries will be left in this discipline due to the
application ofthe Transcendental Philosophy:
Y esta esperanza se funda en el hecho, cada vez más patente, de que los
puntos más obscuros, por menos explorados o más difíciles para la ciencia,
son precisamente aquellos que la ponen eli contacto con la filosofia del
espíritu humano. De esta filosofia transcendental y crítica es de donde
únicamente puede venir la luz: hágase un esfuerzo, remuévase el obstáculo, y
ella se derramará a torrentes, inundando con claridad igual todo el cuerpo
de la ciencia (p. 2)7.
Next come the presentation of one of those obscure points, which Rey
names imaginarismo. In his own words: el imaginarismo es el "scandalum
mathematicum ". And he concludes:
Es necesaria una teoría transcendental del imaginarismo, que salve todas las
contradicciones y dé a la ciencia matemática aquel esplendor e integridad a
que tiene derecho con mejores títulos que ninguna otra ciencia8•
A historical survey of the theoretical development of imaginary quantities follows
in sections JII and IV. According to it, purely mathematical reasons are not enough
basis for the understanding of imaginaries: Sorne think ofthem as the expression of
the impossibility of solving certain equations, Condillac is an idéologue who thinks
that they are senseless signs [Condillac 1769], sorne others think of them as logical
7 And this hope is based on the very patent fact that obscure points [in Mathematics], possibly
because they are less explored or more difficult, are those establishing contact with the philosophy
ofthe human soul. Light will be thrown only rrom this transcendental and critica! philosophy:
Make en effort, remove the obstacle, and it will torrentially spread, tlooding with clarity ali the
body of science.
8 There is a need of a transcendental theory of imaginarism to overcome ali contradictions, and to
endow mathematical science with the splendour and integrity it deserves, for it has better right to
them than any other science.
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symbols, Hoene Wronski (1778-1853) is described as trying to submerge imaginary
quantities in the realm of infinity, and finally Buée is credited to have discovered
the true interpretation of imaginary quantities as a representation of
perpendicularity. Other authors who favour this viewpoint are cited, like Joseph
Gergonne (1771-1859), Warren and Peacock, M. F. Valles, who appears twice as
the author of an Essai sur la philosophie du Calcul (he was also the author of Des
formes imaginaires en Algebre: Leur interpretation en abstrait et en concret
(Gauthier-Villars, Paris, 1869 and 1876)) and C. V. Mourey, who published in Paris
La vraie théorie des quantités négatives et des quantités prétendues imaginaires
the year 1818. According to Rey he introduced the word versar as a name for the
argument of a complex number. Two more authors, opposed to this advance, are
also considered: Charles Renouvier (1815-1903) [Renouvier 1950], and Augustin
Cournot ( 1801-1877).
Sections V and VI are devoted to a reflection on the nature of Algebra and
its ability to cope with the representation of ideas more complex than Arithmetic.
Rey decidedly asks and answers this question in a single paragraph:
¿Puede haber afección o relación geométrica que no sea representable por
el Álgebra? Yo digo resueltamente que no: y presento como demostración el
contenido material de este libro ... 9 ,
and goes on to assert in p. 6 that according to Pierre Simon de Laplace (1749-
1824) "the Metaphysics of these sciences has not yet been made", and that
algebraic notation is endowed with a better analytical ability than purely
arithmetical manipulations:
Como lengua, alcanza el Álgebra esa superioridad a la que no tiene derecho
como ciencia limitada a números10•
Finally, and here the ldéologie is rather apparent, he states that Algebra is more
than a universal language for relationships and properties of numbers, and its
universality is not merely a philological one: It is objective, and Algebra deals with
matters that cannot be studied only by way of Arithmetic or Geometry, either
isolated or jointly considered.
9 Are there any geometrical relationships or affections not describable through Algebra? Definitely,
1 say no: And the proofis the material content ofthis book ... [to Rey, ajfection means modification
of the meaning of a arithmetical entity by sorne additional information, e.g. the minus sign in front
of a positive number changes the sense of its measure on the line]
'ºAs a language, Algebra attains a superiority to which it has no right as a science limited to
numbers. This is taken straight away from p. 25 ofBuée's Mémoire.
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In order to introduce the kantian scheme, two more sections, VII and VIII,
follow where doing mathematics is postulated asan innate quality ofhuman mind:
... este conocimiento es formado por la virtualidad propia de nuestro espíritu,
sin deber a la experiencia más que la ocasionalidad de su excitación; verdad
contra la cual tanto se rebela el espíritu sensualista y empírico que
heredaron de la Filosofia del siglo anterior los matemáticos, más que ningún
otro linaje de pensadores. 11
According to Rey our minds are more mathematical than we know, because our
mental structures rely more on the logical truth of judgements and propositions, in
opposition to empírica! science, which proceeds slowly through experience. This
has so reduced an extension that the universality and necessity of mathematical
propositions overflows it. Transcendental Philosophy, when trying to explain this
special nature of mathematical knowledge, is forced to recognise the purely
subjective and formal nature of the grounds of geometrical and arithmetical
intuitions: space and time. These two concepts do not derive from experience, they
are prior to it and are transcendental conditions thereof.
The intuition of space, once · stripped of any phenomenic or sensible
element, is that of a simultaneous, infinite, and homogeneous capacity where
nothing is predetermined. On the other hand, the pure intuition of time is that of an
infinite sequence of essentially succesive and transitory moments. Rey notes that
En esta serie están determinadas las cosas como numerables: la numeración
es imposible sin la síntesis sucesiva de la unidad consigo misma ... 12
The idea of synthesis of the unit with itself is clearly a translation of
addition, and is found at the basis of axiomatic descriptions of natural numbers,
like the well known Peano formulation. Rey uses this conception along Teoría, in a
picturesque language where e.g. the fact that 1x 1=1 is described like "the
barrenness of unity under production" and other similar expressions.
A succint historical description of Algebra is found in section IX, where
after considering Descartes as the discoverer of Scientific Algebra and Euclid as
the first instance where Arithmetic .and Geometry are sister disciplines, he
11 This knowledge [the mathematical one] is built by the innate powers of our spirit, it owes to
experience only the occasion to excite them; and against this truth rebels the sensualist and
empiricist spirit inherited by mathematicians - more than by any other scientific lineage- from last
Century's Philosophy.
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concludes with the above quoted phrase of Pascal, which appears here for the first
time in the book (p. 16).
Section X starts by asking the question of how could the problem of
imaginary numbers be settled in his Pascalian programme. Here is the solution:
La teoría geométrica de la posición, o analysis sitús, que era el gran
desiderátum de Leibnitz{. . .}l/egará a convertirse en una teoría algébrica
ordinaria, desde el momento en que se nos den signos para todas las
posiciones de las rectas en el plano indefinido o en el espacio, con tal que
estos signos no sean arbitrarios, sino engendrados por el mismo cálculo
como resultados algorítmicos necesarios de las teorías de los números. 13
Therefore, a theory of signs must be built, and the basics thereof must take into
account the concept of quality in order to represent every "affection" or direction:
The road is now free for the appearance of Transcendental Logic and the kantian
categories. This is the aim of the last two sections, XII and XIII, where a rather
hard vindication of Metaphysics as the root of Mathematics is presented. The
following excerpt makes it clear (p. 20):
¡Horror a la Metafisica los matemáticos, cuando ellos sin quererlo, y sin
saberlo, son los primeros metafisicos! ¡Cuando las ideas de espacio, tiempo,
movimiento, nada, é infinito aparecen a cada momento como enredadas
entre sus teorías, ó siendo la trama de sus cálculos/14
And he ends: The best mathematicians praised by History were deep
metaphysicists as well, and it is certain that Gottfried Leibnitz (1646-1716), René
Descartes (1596-1650), Isaac Newton (1642-1727), Pascal or Leonhard Euler
(1707-1783) would not have been so higher mathematicians if their philosophical
insight had not been as deep as it was. Moreover: Who ignores that Kant was a
mathematician before becoming a criticist? And the same can be said of Reid,
Georg Hegel (1770-1831) and Karl Krause ( 1781-1832). And in p. 22:
12 In this sequence things are enumerably determined: enumeration is impossible without the
succesive synthesis ofunit with itself ... Synthesis is a very frequent word in Teoría, where it is used
with the general meaning of"sum, combination, ... "
13 Geometrical position theory or analysis sitüs, a desideratum ofLeibnitz [ ... ] will become an
ordinary algebraic theory from the moment we are given signs for every possible direction in the
plane or in the space, taking care that these signs are not arbitrarily chosen, but derived
algorithmically from computations carried on with numbers.
14 How come that mathematicians abhorr Metaphysics, when they are -without wanting or knowing
it- the very first metaphysicists! Moreover, when ideas like space, time, force, movement, nothing,
and infinity appear like braided in their calculations!
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La crítica kantiana es un admirable trabajo matemático15
Finally, a defence of the kantian critique based on its mathematical value is
expounded and the Tntroduction closes with a wish for a long and fruitful alliance
between Philosophy and Mathematics.
6. 2. Sorne aspects in the body of Teoría
The contents in Book 1 are very close to those ofBuée's Mémoire, and Rey's
goal is to state the definitions and results as stemming in a natural way from the
intellectual framework of categories.
According to Rey and his inspirer Buée, imaginary quantities must be
understood through an application of the quality concept. Here, quality is an
addition to the quantity itself, considered as a modifying attribute or adjective, in
such a way that the arithmetical property of number known as quantity is
completed with a geometrical viewpoint as well. The simplest case is that of
negative numbers, where the minus sign indicates reversa) of the sense used to
represent the original numbers as segments emanating from a definite point in a
line. Observe that an important difference between the nomenclature used by Buée
or Rey and contemporary use is that fine was used to mean a finite segment, and
direction meant a choice in the sense used to define the segment once a fixed end
was given. The status of negative quantities was thoroughly studied by the end of
the 18th Century, e.g in the long time forgotten memory of Wessel published in
Copenhagen, and in 1802 the philosopher and later freemason Krause, quoted by
Rey in pp. 22 and 32 of Teoría, read an Habilitationsschrift in Jena dealing with
this topic with the title De philosophiae et matheseos notione et earum intima
coniunctione. An interesting account on the romantic character of the subject can
be read in [Dhombres 2003]. Krause, who was to become an influential
philosopher in sorne intellectual Spanish and Latín American circles during the
I 9lh Century, wrote a second essay in 1804 on the relationship between Philosophy
and Mathematics under the title Grundlagen eines philosophischen Systems der
Mathematik. The sound Jogical foundations were established by Carl Friedrich
Gauss (1770-1855) [Gauss 1831], and the field structure of the complex numbers,
by Hamilton around the same time.
15 Kanfs critique is an admirable mathematical piece.
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limitación. En la teoría de la potencialidad, ó sea de la graduación, está el
germen de la teoría cualitativa20•
This excursion into the symbolic quality of ~ and its possible connection
with algebraic, nonmetaphysical questions, is the core of Rey's argumentation and
reveals a philological, serniotic and idéologique origin in his research.
Now to the construction of the square root of -1. Rey argues that the square
root of -1 cannot be either -1 or + 1, for según la doctrina comun y verdadera de
los algebristas21 , or rule of signs, the square of any of them will be + 1. Therefore
-in the same way that Wessel and Buée had indicated sixty years before- the root
rnust lie in sorne direction different from the one defined by -1 and + 1, and he
goes on in his rhetoric style:
... la posición indirecta, que corresponde a la raíz ~, debe
constituir un grado tal de evolución cualitativa respecto de la unidad
positiva, que repetido este grado en igual razón y manera progresiva,
constituya a su vez la segunda potencia -1 ... 22
Note in this text the inadequate usage of~, in an instance where the
expression "the square root of-1 ", rather than the symbol, should have been used.
This error shows that Rey had sorne difficulties in distinguishing concepts when he
tried to put them in mathernatical forrnulation. In any case, the rationale in this
paragraph is that the positive unity + 1 suffers a qualitative evolution that
succesively takes it into other directions before arriving at the negative part of the
axis. Of course Rey does not rnention it explicitly, but it is plain that he is thinking
of rotating the positive axis around the origin without leaving the plane, an idea
dueto Car! Friedrich Gauss (1777-1855) [Gauss 1831]. According to this picture,
the square root of -1 will be obtained after sorne particular evolution, and upon
iteration of this same evolution on the square root, the square - 1 will be obtained
20 ln the previous chapter l considered the sy~bol ~as a sign of limitation or perfect neutrality
between the positive and the negative affections, oras the most adequate expression ofthe
maximum degree ofindifference towards these fundamental directions. Nevertheless, the radical
form ofthe symbol reveals a potential (with the form ofa power) algorithmic origin, and is the
symbol of a immense theory, from which the three typical moments +,-,±~are particular
determinations corresponding to the concepts of reality, negation, and limitation. The germ ofthe
qualitative theory is in the theory ofpotentiality, i.e. of graduation (raising to powers).
21 •.• according to the common and true doctrine ofalgebrists ...
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axis will represent a particular quality B, where the origin O separates the ideas of
"positively having it" or "negatively having it". Thus, if R(A, B) means that a point
A has been chosen in the positive half of the B axis, R(A,-.B) will mean that A has
been chosen in the negative half. For any direction other than B, the actual
situation is -.R(A, B), so necessarily the relationship
R(A,-.B) *- -.R(A,B)
will hold. Nevertheless, degrees in being or not being B do actually exist19, and
they are represented by ali directions defined by the origin and the points in a circle
passing through O and whose centre C is such that R(C,B). Therefore, the only
direction whose representative line has no common point with the circle other than
the origin will be the perpendicular to the axis through O, and it will represent
complete indifference with respect to the quality B. This leads Rey to the
provisional and arbitrary adoption of Has a sign of quality, on equal foot with +
or -, denoting perpendicularity as the representation of total indifference towards
the quality B.
The above idea, though an appealing one, is of little use in everyday
mathematical practice. Thus Rey embarks in a justification of this interpretation by
combining it with the classical problem of extracting the square root of a negative
number. In his own words (Teoría, p. 51 ):
En el anterior capítulo he considerado el símbolo H como un signo de
limitación ó de neutralidad perfecta, entre la afección positiva y la negativa,
ó como la expresión más propia de un grado máximo de indiferencia
respecto de aquellas direcciones fundamentales. Sin embargo, por su forma
radical revela el signo H un origen algorítmico potencial, y simboliza la
totalidad de una teoría inmensa, de la cual no son sino determinaciones
particulares los tres momentos típicos representados por los signos
+, -, ± H correspondientes a los tres conceptos, realidad. negación y
19 In a certain sense, Fuzzy Logic was been used many years before it carne into being.
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limitación. En la teoría de la potencialidad, ó sea de la graduación, está el
germen de la teoría cualitativa20•
This excursion into the symbolic quality of ~ and its possible connection
with algebraic, nonrnetaphysical questions, is the core of Rey's argurnentation and
reveals a philological, serniotic and idéologique origin in his research.
Now to the construction of the square root of -1. Rey argues that the square
root of - 1 cannot be either -1 or + 1 , for según la doctrina comun y verdadera de
los algebristas21, or rule of signs, the square of any of thern will be + l . Therefore
- in the same way that Wessel and Buée had indicated sixty years before- the root
rnust líe in sorne direction different frorn the one defined by - 1 and + 1, and he
goes on in his rhetoric style:
... la posición indirecta, que corresponde a la raíz H, debe
constituir un grado tal de evolución cualitativa respecto de la unidad
positiva, que repetido este grado en igual razón y manera progresiva,
constituya a su vez la segunda potencia - 1 ... 22
Note in this text the inadequate usage ofH, in an instance where the
expression "the square root of-1 ", rather than the syrnbol, should have been used.
This error shows that Rey had sorne difficulties in distinguishing concepts when he
tried to put thern in mathernatical formulation. In any case, the rationale in this
paragraph is that the positive unity + 1 suffers a qualitative evolution that
succesively takes it into other directions before arriving at the negative part of the
axis. Of course Rey does not rnention it explicitly, but it is plain that he is thinking
of rotating the positive axis around the origin without leaving the plane, an idea
dueto Carl Friedrich Gauss (1777-1855) [Gauss 1831]. According to this picture,
the square root of - 1 will be obtained after sorne particular evolution, and upon
iteration of this same evolution on the square root, the square - l will be obtained
20 In the previous chapter l considered the sy~bol ~as a sign of limitation or perfect neutrality
between the positive and the negative affections, oras the most adequate expression ofthe
maximum degree ofindifference towards these fundamental directions. Nevertheless, the radical
form of the symbol reveals a potential (with the form of a power) algorithmic origin, and is the
symbol of a immense theory, from which the three typical moments +,-,±~are particular
determinations corresponding to the concepts ofrea/ity, negation, and limitation. The germ ofthe
qualitative theory is in the theory ofpotentiality, i.e. ofgraduation (raising to powers).
21 ••• according to the common and true doctrine of algebrists ...
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in a supposedly natural way. But the only direction having this property is that at
right angles, and he concludes:
Luego la perpendicular representa la raíz segunda de -1, y el símbolo
radical H es a su vez genuina expresión de la perpendicularidad. 23
Next, the above deduction is performed in mathematical notation, with a
pretension of generality: Let a be a quantity representing the square root of - a ,
and let p be the coefficient or particular evolution that acts on a far it to become
the square root of -a, i.e. p(a) = pa = SQRT(-a). Then,
... es claro que, repitiendo esta modificación sobre la raíz, quedará
constituida la segunda potencia - a , y se tendrá ... 24
In formulas, this is written as:
p(p(a)) = p(pa) = p 2 (a) = p 2a =-a
and on dividing the last equation through a the desired result will appear:
p 2 = -1 , or equivalently, p = ±H
A critica! study of this argument is worth, because there are severa! flaws or
obscure points in it:
• No mention 1s made to the point that a must be a positive quantity.
Otherwise the argument, which amounts to two consecutive rotations of
equal angle, will fail.
• More important is the fact that the above argument will not work for a 7: 1.
In arder to comply with the usual rules of extracting roots, a modification
on the magnitude of a must happen simultaneously to the rotation. Suppose
e.g. that a > 1 . Then, in the first application, the action of p is split into a
rotation and a shrinking a ~ +Fa . Therefore in the second application p
22 . .. the indirect (oblique) position correspondíng to the square root must be obtained by a degree of
qua/ilative evo/u/ion of the positive unity such that, on repeating it, the square number - l will be
obtained ...
23 Therefore the perpendicular represents the square root of - 1, and the radical symbol H is in
tum a genuine expression ofperpendicularity.
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should perform a stretching +Fa --+a and a rotation. In other words, it
would be a different p!
• The equation p 2 = -1 is not an equation involving numbers: it is a purely
formal one, and its correct meaning is that the iteration of sorne operator
acting on a positive quantity yields the opposite thereof, so the algebraic
origin of P. remains hidden in the dark.
The original argument in Buée's Mémoire (p. 28 and 29), on which this
explanation is inspired, is much simpler and less elaborate. Stripped of rhetoric, the
essential of both approaches amounts to applying the numerical idea of
proportional means by extending it to operator-like entities whose models are
rotations of the plane around a fixed point.
Algebraic manipulation of complex numbers is studied in Books JI, III, and
IV of Teoría. The names used by Rey are: Synthesis far addition, production far
multiplication, and graduation far raising to powers. Apart from this nomenclature,
Rey presents and explains the componentwise addition and the multiplicative rule
as syncategorematic operations, thus indicating that they consider jointly both
quantitative and qualitative aspects of complex numbers. Already in 1831 Cauchy
had studied complex numbers as ordered pairs [Cauchy 1831] [Pérez-Fernández
1999]. Apart from this philosophical touch, ali tapies in Books II and 111 are
commonplace and are classically considered. Book IV, dealing with graduation or
raising to a power, is much more interesting.
6. 3. Book IV and Rey's mathematical abilities
The reader can observe in Book IV the real mathematical background of Rey
and to which extent he had understood it. The Book starts with sorne general ideas
on graduation and how it works far positive quantities, or "quantitative graduation"
and far negative ones, or "qualitative graduation". lndeed the difference is that
qualitative graduation displays an oscillating behaviour. On extending graduation
to imaginary quantities (p. 169 ff.) the periodic behaviour of powers of P. is
established after a lengthy preamble where the validity of the operation according
24 .•. it is clear that by repeating this modification upon the root, - a, the square, will be
constituted ...
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to Rey's original plan is acknowledged. A few pages are devoted to involution or
root extraction, as the operation inverse to graduation.
1
ln p. 179, in order to find the binomial form of [.J=-1 t Rey <loes not hesitate
to use an infinite number of times the binomial or Newton series expansion for
in a first instance for the global expression, and then substituting the corresponding
expansion for every appearance of the expression (1 - .J=-1 Y . The reader is forced
to conclude that to Rey the authority argument represented by invoking the name
ofNewton dominates every other consideration on convergence, actual infinities or
any other mathematical ideas. This is nota case ofMetaphysics: he is simply using
well established rules, and this is reason enough to accept them without further
thought. This was all right in Buée's times, but it should be reminded that Augustin
Cauchy ( 1789-1857) had already attempted to introduce rigour in these matters
forty years before.
Cyclotomic polynomials are also extensively considered, and a trigonometric
theory of imaginaries is presented (p. 233 ff.), a topic addressed by ali authors
studying complex quantities. lt is found in Buée, and had been common fare in
academic sessions, as shown e.g. in [Svanberg 1812]. Graphical constructions also
deserve a lot of attention, and then Chapter VII on infinite graduation of imaginary
quantities starts.
Under the rather mysterious title Jnevolubilidad esencial de la unidad bajo el
concepto de cantidacf5 Rey explains that the unit 1 remains the same regardless of
the exponent -even an infinite one- when graduation (l)"' is performed. Now Rey
asserts that if unity is perturbed by the addition of an infinitesimal quantity, infinite
graduation will fruitfully evolve, so he proceeds on to produce the marvellous
formula [Euler 1748, Chapter VII] :
(l +μ-)"" = l+-x-+--x-+---x -+ 00 μ 002 μ 2 003 μ 3 •
00 1 00 1 . 2 002 l . 2 . 3 003 .. .
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where working a la Euler infinite quantities cancel, and upon setting μ = 1 the
number e is obtained. Then the fundamental property
is established and a remark on the practica! computation of logarithms is presented.
In p. 203 a new version of the mysterious heading appears: Inevolubilidad
de la unidad positiva bajo el concepto de calidad. Número g_ cualitativo26. lt is
natural that Rey should try to eliminate such an undesirable non evolutionary
behaviour by adapting the above reasoning for the case μ = ~ . In the same
mood he defines the qualitative number :
E= (l+ P--J00 = 1+~-- --1 --~-+ 1 + ...
00 1 1·2 1· 2·3 1·2·3·4
And when it comes to find the binomial expression for E, he wishfully writes:
En esta serie, cuya estructura es ya conocida [ . .} es posible reconocer la
convergencia de los términos reales cuya suma es a, y de los coeficientes de
los imaginarios cuya suma es /3 , a un punto común en que ambas sumas se
hacen iguales; de suerte que en el límite en que esto se verifica es un punto
de tal naturaleza, que respecto de él se tiene a = /3 y la expresión
sumatoria de la serie infinita es ( 1 + :1J00
=a+ a~ 27•
Of course this is not true. Rey's phrasing "it is possible to recognise ... " would ha ve
better been applied to recognise in E the exponential
e..H = o.s4o302 ... + o.8414 71 ... P,
25 The essential lack of evolution of unity under the concept of quantity.
26 The lack of evolution of unity under the concept of quality. Qualitative version of e.
27 In this expansion it is possible to recognise the convergence ofthe real terms and ofthe
coefficients ofthe imaginary ones to a common point where both sums become equal...
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the unit-like complex number 18 =1 with unit module and argument one radian. This
spoi ls the beautiful symmetry of the theory to be developed from Rey's expected
value E =~.J=I.
The final paragraphs of the book are equally disappointing. The ideas of
Buée and Argand suggest immediately the possibility of extending to space the
theory ofperpendicularity: Is it possible to representa line perpendicular to a given
plane by means of sorne particular complex number a+ b.J=I? Many years
before, in 1813, Frarn¡:ois Servois (1768-1847) had tried to answer the question in
the negative, establishing a precedent of the theory of quatemions later developed
by Hamilton. According to Rey, the double exponential (.J=Jr <loes the job, and
he justifies it in this cryptic paragraph (p. 284):
Que tal debe ser la evolución, se infiere del efecto natural de los
exponentes reales sobre una base imaginaria. Si el exponente positivo
le imprime una moción progresiva, el exponente cero la coloca sobre
el radio, y el negativo la hace retrogradar por bajo de esta posición;
el imaginario la elevará desde este origen de los arcos por un arco
que puede llamarse imaginario como perpendicular a los arcos
positivo y negativo.28
Therefore the six quantities,
{ + 1,-1,+.J=l,-.J=l,+(.J=J)~ ,-(.J=J)~ }
are postulated as a system from which Space Geometry could be derived. Needless
to say, Rey never attempts to prove that (.J=Jr can be expressed in the
forma+ b.J=I : without further check, he simply thinks of it as a new species of
complex or imaginary quantity. And so did sorne of his followers (see below), even
ifthey knew that (.J=Jr is a real number.
28 That this must be the evolution can be inferred from the natural effect of real exponents upon an
imaginary basis. Ifa positive exponent impresses a progressive motion, a null one will leave it on
the radius, anda negative one will impress a retrograde movement under this position; an imaginary
one will elevate it from this origin of ali ares following an are that can be called imaginary because
it is perpendicular to the positive and the imaginary ares.
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In p. 291, just before the translation ofKant and the final glossary, the author
presents a synthesis of his work in three mottos and their attribution to various
intellectual realms:
l. The symbol ~ is a sign ofperpendicularity (Buée)29.
2. Numbers imitate space, though their natures are so different (Pascal).
3. The tableau of categories indicates every moment of any projected
speculative, even providing its ordering and regime (Kant)3º.
And then, to where they belong:
• The first one is a pure mathematical thought.
• The second belongs to Mathematical Philosophy.
• The third, to Transcendental Philosophy.
7. Sorne epigons of Rey
The theories of Rey enjoyed a certain popularity in the following years, and
severa) mathematicians incorporated sorne of his ideas in their texts, as stated in
the survey [García de Galdeano 1899) and in the words of Julio Rey Pastor (1888-
1962), one of the founders of contemporary Spanish Mathematics [Rey Pastor
1915). Here sorne of those followers are presented, even at the risk of forgetting
more than one:
7.1. Rochano
José Antonio Rochano Alemany, who was a Professor in Granada, appears as
a very devoted follower of Rey, with whom he had been apparently acquainted. In
1870 he published his Elementos de Algebra [Rochano 1870), a text which is a
mixture of a close copy of Teoría and more common algebraic topics, like GCDs
and other usual concepts. Rochano cites Rey explicitly severa) times along the text,
29 Ainsi ~ est le signe de la perpendicularité, dont... [Buée 1806], p. 28.
30 Rey offers a compressed version ofthe original sentence ofSection 3, Paragraph 11 in Des
transzendentales Leiifadens der Entdeckung aller reinen Verstandbegriffe: ,,Denn daj3 diese Taje/
im theoretischen Teile der Philosophie ungemein dienlich, ja unentbehrlich sei, den Plan zum
Ganzen einer Wissenschafi. sofern sie auf Begriffen a priori beruht, vollstandig zu entwerfen, und
sie mathematisch nach bestimmten Prinzipien abzuteilen, erhellt schon von selbst daraus, daj3
gedachte Ta/el al/e Elementarbegriffe des Verstandes vollstandig,ja selbst die Form eines Systems
derselben im menschlichen Verstande enthalt, folglich auf a/le Momente einer vorhabenden
spekulativen Wissenschafi, ja sogar ihre Ordnung. Anweisung gibt, wie ich denn auch davon
anderwarts eine Probe gegeben habe ".
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and emphasises in the most polemic aspects presented in Teoría, such as spherical
imaginaries, but goes no further than Rey. During this study no other mathematical
activity of Rochano has been found. Next, a sample of his writing (p. 91) is shown,
the introduction to a comment on the fact that h x N is a real number:
Consecuencia importante. Luego el producto de dos imaginarios es REAL.
¿Cómo se ha obrado este milagro? ¿Con que dos cantidades simbólicas,
quiméricas, imposibles, visionarias, combinadas por la vía de la producción,
dan una cantidad que ya no es quimérica y visionaria? ... 31
7.2. Domínguez Hervella
The Navy officer Modesto Domínguez Hervella, published sorne manuscript
lecture notes around 1 870 under the title Elementos de Geometría analítica, which
were later put in print with the same title in 1879 [Domínguez 1879]. A copy of
this last one has been used. The book starts with a quotation from the Introduction
of Teoría, and many illustrations are clearly inspired in the original ones of Rey's
text. Hervella is much more technical than Rey or Rochano, but he still sticks to
the use of (.J=IJ for the study of space Geometry, even when he is aware of the
non-imaginary nature of this quantity. In a report of the Real Academia de
Ciencias included in the book32, an anonymous reviewer reproaches the author for
not adopting Hamilton's system instead of the old fashioned, complicated, and
erroneous double imaginary.
7.3. Navarro
Luciano Navarro Izquierdo, a Professor at Salamanca, published in 1874 a
Tratado de Geometría elemental y Trigonometría rectilínea y esférica, a book that
had two more editions in 1883 and 1887. For this study a copy of the second
edition has been used [Navarro 1883]. This author uses freely ideas taken from
Rey, and although he <loes not explicitly cite him, he employs complex numbers in
routine trigonometric computations, as in deriving the sine and cosine of the
double angle. Moreover, he was the author of a Tratado de Aritmética y Algebra
31 An important consequence: How did this miracle happen? How come that the combination
through production oftwo symbolic, chimerical, impossible, visionary quantities can yield a
~uantity that is no more chimerical and visionary? ...
3 A positive report opened the <loor to automatically selling three hundred copies for public
libraries ali over Spain.
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appeared in 1875, and a report in I 886 on the procedure of Bemardino de Sena for
the approximation of square roots. Here is a sample of Navarro's style in
Geometría (p. 23), where he explains the concept of direction considered as a
quality:
Puesto que la cualidad de la cantidad no es otra cosa que su modo de ser u
obrar con relación a la tomada por unidad, es evidente que toda recta, cuya
dirección sea la misma que la de la unidad, es de la misma cualidad que
ésta, siendo de cualidad distinta si su dirección es diferente: la cualidad de
una recta no es otra cosa, pues, que la dirección de la misma respecto de la
unidad. 33
Then he proceeds with the same formal calculation yielding ~as the mark of
perpendicularity.
7.4. Lubelza and Pérez de Muñoz
Next come the mining engineer Joaquín Lubelza, who was a Professor at
the Escuela de Minas in Madrid. He was the author of a nearly modem Calculus
text published in 1905 [Lubelza 1905) with the title Cálculo Infinitesimal, a text he
had previously distributed in lecture notes forrn around 190 l. Quite interestingly,
the Introduction deals with quaternions, and Rey is cited as having discovered the
syncategorematic character of vector sums (instead of vector, the word rector is
used throughout). Lubelza wrote a booklet on Mechanics in 1901, as well as an
article in 1902 on centrifuga! forces.
To end with Rey's epigons, another mining engineer, Ramón Pérez de
Muñoz published in 1914 his Elementos de Cálculo Infinitesimal, [Pérez de Muñoz
1914) where the Introduction on quaternions was changed into a survey of
Cantor's set theory, the name vector was adopted and quatemions were relegated
to a small Appendix. Significantly enough, no mention to Rey is made in this
book.
33 Given that the quality of a quantity is nothing but the way it behaves in relation to the one taken
as unity, it is obvious that every segment whose direction is the same as that ofunity, has the same
quality as this one ... : the quality of a segment is its direction relative to unity [This is a complicated
and obscure paragraph, much in the style ofRey].
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8. Conclusions
Jt is known (see e.g. [Pacheco et al. 2005]) that during the first half of the
19th Century the study, teaching and dissemination of Mathematics in Spain was by
no means an organised activity, and only a handful of isolated characters worked in
this field of knowledge, mostly in military schools or in the secondary teaching
system. Advances occurring in Europe arrived late and often through second-hand
reports, and it is no surprise that Spanish authors stuck to certain attitudes
preferring either to follow ¡ gth Century Mathematics in its most practica! aspects,
or to study sorne problems half way between Mathematics and Philosophy which
at first sight did not require much mathematical ability.
In this paper one such case has been considered: The reception of
metaphysical explanations of complex numbers and how they influenced Spanish
Mathematics for over fifty years. The foundational crisis of the second half of the
¡9th Century had little impact in Spain [Pacheco 1991], very possibly dueto the
poor technological development of the country, for it must be reminded that a great
<leal of problems in the foundations of Mathematics had deep roots in the
flourishing of severa! classes of machines [Pacheco and Femández 2005]. One of
the outcomes of the crisis was the exigence of rigour in both senses addressed in
the lntroduction in order to consistently study the mathematical models underlying
the functioning of the new technologies. The work of Rey tried to be rigorous in
the metaphysical and in the logical side, but failed in the application of
"mathematical" rigour, as shown in Nº 6. Thus, it is no wonder that Rey's book
could enjoy a lasting popularity well above his mathematical value, as shown in
[García de Galdeano 1899] and in [Rey Pastor 1915]. This seems to be a case
where the distinction pointed out between history and heritage in [GrattanGuinness
2004] is perfectly applicable. Nevertheless, the merit of Rey must be
recognised when facing a difficult and hard problem with the na"ive appreciation
that it could be solved from scratch through the straightforward application of
purely philosophical techniques. His courage deserves a place in the history of
Spanish Mathematics of the 19th Century, even if the final result was not as
brilliant as expected and many a problem remained unsolved.
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